Securing broadcast against dishonest receivers

Consider a sender, Alice, who wants to transmit private messages to two receivers, Bob and Calvin, using unreliable wireless broadcast transmissions and short public feedback from Bob and Calvin. In [1], we assumed that Bob and Calvin provide honest feedback, and characterized the secure capacity region of the private messages under the requirement that Bob and Calvin do not learn each other´s message. In this paper, we assume that Bob (or Calvin) may provide dishonest feedback; or even control the input message distributions, as is commonly assumed in cryptography literature. We characterize the capacity region in the case of dishonest adversaries, as well as an achievable region for the case when the adversary has complete control on the distribution of the messages. We also design polynomial time protocols for both cases, that rely on the use of coding techniques to mix and secure the private messages. As a side result, we define an extended notion of semantic security for this problem and using a similar approach to [2], we show the equivalence of different security notions.